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| Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.) | 
| Ref | Expression | 
|---|---|
| int0 | ⊢ ∩ ∅ = V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ral0 4513 | . . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥 | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 3 | 2 | elint2 4953 | . . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥) | 
| 4 | 1, 3 | mpbir 231 | . . 3 ⊢ 𝑦 ∈ ∩ ∅ | 
| 5 | 4, 2 | 2th 264 | . 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V) | 
| 6 | 5 | eqriv 2734 | 1 ⊢ ∩ ∅ = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∅c0 4333 ∩ cint 4946 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-dif 3954 df-nul 4334 df-int 4947 | 
| This theorem is referenced by: unissint 4972 uniintsn 4985 rint0 4988 intex 5344 intnex 5345 oev2 8561 fiint 9366 fiintOLD 9367 elfi2 9454 fi0 9460 cardmin2 10039 00lsp 20979 cmpfi 23416 ptbasfi 23589 fbssint 23846 fclscmp 24038 zarcmplem 33880 rankeq1o 36172 bj-0int 37102 heibor1lem 37816 ipoglb0 48883 mreclat 48886 | 
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